Question

Show \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})=\|\mathbf{u}\|^{2}-\|\mathbf{v}\|^{2}\) for any vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Short answer

This identity uses the distributive property and properties of dot product.

Step-by-step solution

Expand the Expression

First, expand the expression \((\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v})\) using the distributive property of the dot product:\[(\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v}) = \mathbf{u} \cdot \mathbf{u} - \mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v}\]

Simplify Using Properties of Dot Product

Notice that dot product is commutative, meaning \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\). We can simplify the expression:\[\mathbf{u} \cdot \mathbf{u} - \mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v}\]

Relate to Magnitudes

The expression \(\mathbf{u} \cdot \mathbf{u}\) is the square of the magnitude of \(\mathbf{u}\), and \(\mathbf{v} \cdot \mathbf{v}\) is the square of the magnitude of \(\mathbf{v}\). Thus, we have:\[\mathbf{u} \cdot \mathbf{u} = \|\mathbf{u}\|^2\] \[\mathbf{v} \cdot \mathbf{v} = \|\mathbf{v}\|^2\] Substitute back into the expression:\[\|\mathbf{u}\|^2 - \|\mathbf{v}\|^2\]

Key concepts

Vector Magnitude

When discussing vectors, one important concept is the magnitude. The magnitude of a vector \( \mathbf{u} \) is essentially its "length" or "size". It is denoted by \|\mathbf{u}\|\. This is a crucial aspect to understand in vector operations.
To compute it, we use the dot product of the vector with itself: \mathbf{u} \cdot \mathbf{u} = \|\mathbf{u}\|^2\.
This result is the square of the vector's magnitude. Essentially, the dot product of a vector with itself gives us a measure of how "long" it is in n-dimensional space.

• This is computed as the sum of the squares of its components, much like the Pythagorean theorem.
• The square root of this sum gives the actual magnitude or "length".

Understanding vector magnitudes helps in grasping how vectors interact in equations.

Distributive Property

The distributive property is vital in vector algebra, especially when dealing with dot products. This property states that the dot product is distributive over addition.
For vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), one can express it as: \( (\mathbf{a}+\mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c} \).
This allows us to "distribute" the dot product over addition, just as in arithmetic with numbers.

• The property simplifies complex expressions by separating sums within dot product calculations.
• It enables the simplification of vector equations, making it easier to find solutions.

In the original exercise, it helps expand the expression, making simplification possible.

Commutative Property

The commutative property of the dot product is a straightforward yet powerful rule. It states that the dot product will yield the same result regardless of order: \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
This means factors in a dot product can be switched without affecting the outcome. It is a foundational aspect of vector manipulation.
By using it, we can often discover simplifications in vector algebra.

• The original exercise uses this when simplifying \( \mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{u} \).
• The commutative property helps show the equivalence between different forms of the expression.

Remembering this property can often give a clearer picture of how vectors relate and interact.

Vector Algebra

Vector algebra is a branch of mathematics focusing on vectors' operations. This includes addition, subtraction, dot and cross products. These operations allow us to analyze and solve problems in multiple dimensions.
Key elements include understanding both the direction and magnitude of vectors, making them unique compared to single-dimensional numbers.
In particular, algebraic properties, such as commutative and distributive, evolve when applied to vectors.

• These make calculations in physics, engineering, and computer graphics more streamlined and understandable.
• The original exercise showcases these properties with dot product manipulations, highlighting their utility.

By applying vector algebra, you can break down complex multi-dimensional problems into manageable calculations.